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Tauberian and Abelian Theorems for Long-range Dependent Random Fields

Author

Listed:
  • Nikolai Leonenko

    (Cardiff University)

  • Andriy Olenko

    (La Trobe University)

Abstract

This paper surveys Abelian and Tauberian theorems for long-range dependent random fields. We describe a framework for asymptotic behaviour of covariance functions or variances of averaged functionals of random fields at infinity and spectral densities at zero. The use of the theorems and their limitations are demonstrated through applications to some new and less-known examples of covariance functions of long-range dependent random fields.

Suggested Citation

  • Nikolai Leonenko & Andriy Olenko, 2013. "Tauberian and Abelian Theorems for Long-range Dependent Random Fields," Methodology and Computing in Applied Probability, Springer, vol. 15(4), pages 715-742, December.
    • Handle: RePEc:spr:metcap:v:15:y:2013:i:4:d:10.1007_s11009-012-9276-9
    DOI: 10.1007/s11009-012-9276-9
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    References listed on IDEAS

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    1. Frédéric Lavancier, 2007. "Invariance principles for non-isotropic long memory random fields," Statistical Inference for Stochastic Processes, Springer, vol. 10(3), pages 255-282, October.
    2. Paul Doukhan & Patrice Bertail & Philippe Soulier, 2006. "Dependence in Probability and Statistics," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-00268232, HAL.
    3. Anderson, Dale N., 1992. "A multivariate Linnik distribution," Statistics & Probability Letters, Elsevier, vol. 14(4), pages 333-336, July.
    4. Paul Doukhan & Patrice Bertail & Philippe Soulier, 2006. "Dependence in Probability and Statistics," Post-Print hal-00268232, HAL.
    5. Beran, Jan & Ghosh, Sucharita & Schell, Dieter, 2009. "On least squares estimation for long-memory lattice processes," Journal of Multivariate Analysis, Elsevier, vol. 100(10), pages 2178-2194, November.
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    Cited by:

    1. Rosa Espejo & Nikolai Leonenko & Andriy Olenko & María Ruiz-Medina, 2015. "On a class of minimum contrast estimators for Gegenbauer random fields," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 24(4), pages 657-680, December.
    2. Andriy Olenko & Dareen Omari, 2020. "Reduction Principle for Functionals of Vector Random Fields," Methodology and Computing in Applied Probability, Springer, vol. 22(2), pages 573-598, June.

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